Integrand size = 22, antiderivative size = 296 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {x}{b d}-\frac {a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} (b c-a d)}+\frac {c^{4/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^{4/3} (b c-a d)}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac {c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)} \]
x/b/d+1/3*a^(4/3)*ln(a^(1/3)+b^(1/3)*x)/b^(4/3)/(-a*d+b*c)-1/3*c^(4/3)*ln( c^(1/3)+d^(1/3)*x)/d^(4/3)/(-a*d+b*c)-1/6*a^(4/3)*ln(a^(2/3)-a^(1/3)*b^(1/ 3)*x+b^(2/3)*x^2)/b^(4/3)/(-a*d+b*c)+1/6*c^(4/3)*ln(c^(2/3)-c^(1/3)*d^(1/3 )*x+d^(2/3)*x^2)/d^(4/3)/(-a*d+b*c)-1/3*a^(4/3)*arctan(1/3*(a^(1/3)-2*b^(1 /3)*x)/a^(1/3)*3^(1/2))/b^(4/3)/(-a*d+b*c)*3^(1/2)+1/3*c^(4/3)*arctan(1/3* (c^(1/3)-2*d^(1/3)*x)/c^(1/3)*3^(1/2))/d^(4/3)/(-a*d+b*c)*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.80 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {-\frac {6 a x}{b}+\frac {6 c x}{d}-\frac {2 \sqrt {3} a^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{4/3}}+\frac {2 \sqrt {3} c^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{d^{4/3}}+\frac {2 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}-\frac {2 c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{4/3}}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}+\frac {c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{4/3}}}{6 b c-6 a d} \]
((-6*a*x)/b + (6*c*x)/d - (2*Sqrt[3]*a^(4/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^( 1/3))/Sqrt[3]])/b^(4/3) + (2*Sqrt[3]*c^(4/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^( 1/3))/Sqrt[3]])/d^(4/3) + (2*a^(4/3)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) - ( 2*c^(4/3)*Log[c^(1/3) + d^(1/3)*x])/d^(4/3) - (a^(4/3)*Log[a^(2/3) - a^(1/ 3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3) + (c^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1 /3)*x + d^(2/3)*x^2])/d^(4/3))/(6*b*c - 6*a*d)
Time = 0.48 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {979, 1020, 750, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 979 |
| \(\displaystyle \frac {x}{b d}-\frac {\int \frac {(b c+a d) x^3+a c}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{b d}\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \int \frac {1}{d x^3+c}dx}{b c-a d}-\frac {a^2 d \int \frac {1}{b x^3+a}dx}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a^2 d \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}}{b d}\) |
x/(b*d) - (-((a^2*d*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-((Sq rt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/(b*c - a*d)) + (b*c^2*(Log[c^(1/3) + d^(1/3)*x]/(3*c^(2/3)*d^(1/3)) + (-((Sqrt[3]*ArcT an[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(1/3)) - Log[c^(2/3) - c^(1/3)* d^(1/3)*x + d^(2/3)*x^2]/(2*d^(1/3)))/(3*c^(2/3))))/(b*c - a*d))/(b*d)
3.2.9.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d *(m + n*(p + q) + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x ^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x ]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^n), x], x ] - Simp[(d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b , c, d, e, f, n}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 4.52 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {x}{b d}+\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) c^{2}}{d \left (a d -b c \right )}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a^{2}}{b \left (a d -b c \right )}\) | \(225\) |
| risch | \(\frac {x}{b d}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b \,d^{3}-3 c \,d^{2} a^{2} b^{2}+3 c^{2} d a \,b^{3}-b^{4} c^{3}\right ) \textit {\_Z}^{3}+a^{4} d^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-a^{5} b c \,d^{5}-a \,b^{5} c^{5} d \right ) x +\left (-a^{5} b \,d^{6}+3 d^{5} c \,b^{2} a^{4}-2 d^{4} c^{2} b^{3} a^{3}-2 d^{3} c^{3} b^{4} a^{2}+3 a \,b^{5} c^{4} d^{2}-b^{6} c^{5} d \right ) \textit {\_R}^{4}+\left (-a^{6} d^{6}+a^{5} b c \,d^{5}+a \,b^{5} c^{5} d -b^{6} c^{6}\right ) \textit {\_R} \right )}{3 b d}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{4} a^{3}-3 a^{2} c \,d^{3} b +3 a \,c^{2} d^{2} b^{2}-d \,c^{3} b^{3}\right ) \textit {\_Z}^{3}-c^{4} b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-a^{5} b c \,d^{5}-a \,b^{5} c^{5} d \right ) x +\left (-a^{5} b \,d^{6}+3 d^{5} c \,b^{2} a^{4}-2 d^{4} c^{2} b^{3} a^{3}-2 d^{3} c^{3} b^{4} a^{2}+3 a \,b^{5} c^{4} d^{2}-b^{6} c^{5} d \right ) \textit {\_R}^{4}+\left (-a^{6} d^{6}+a^{5} b c \,d^{5}+a \,b^{5} c^{5} d -b^{6} c^{6}\right ) \textit {\_R} \right )}{3 b d}\) | \(415\) |
x/b/d+(1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))-1/6/d/(c/d)^(2/3)*ln(x^2-(c/d)^ (1/3)*x+(c/d)^(2/3))+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d) ^(1/3)*x-1)))/d*c^2/(a*d-b*c)-(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/( a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arc tan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))/b*a^2/(a*d-b*c)
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.77 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} a d \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 2 \, \sqrt {3} b c \left (\frac {c}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} d x \left (\frac {c}{d}\right )^{\frac {2}{3}} - \sqrt {3} c}{3 \, c}\right ) - a d \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - b c \left (\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right ) + 2 \, a d \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 2 \, b c \left (\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b c - a d\right )} x}{6 \, {\left (b^{2} c d - a b d^{2}\right )}} \]
-1/6*(2*sqrt(3)*a*d*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 2*sqrt(3)*b*c*(c/d)^(1/3)*arctan(1/3*(2*sqrt(3)*d*x*(c/d)^ (2/3) - sqrt(3)*c)/c) - a*d*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b) ^(2/3)) - b*c*(c/d)^(1/3)*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3)) + 2*a*d*( -a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 2*b*c*(c/d)^(1/3)*log(x + (c/d)^(1/3)) - 6*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2)
Timed out. \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.18 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{3} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {3} c^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {a^{2} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {c^{2} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {a^{2} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{3} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {c^{2} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {x}{b d} \]
1/3*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((b^3* c*(a/b)^(1/3) - a*b^2*d*(a/b)^(1/3))*(a/b)^(1/3)) - 1/3*sqrt(3)*c^2*arctan (1/3*sqrt(3)*(2*x - (c/d)^(1/3))/(c/d)^(1/3))/((b*c*d^2*(c/d)^(1/3) - a*d^ 3*(c/d)^(1/3))*(c/d)^(1/3)) - 1/6*a^2*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3 ))/(b^3*c*(a/b)^(2/3) - a*b^2*d*(a/b)^(2/3)) + 1/6*c^2*log(x^2 - x*(c/d)^( 1/3) + (c/d)^(2/3))/(b*c*d^2*(c/d)^(2/3) - a*d^3*(c/d)^(2/3)) + 1/3*a^2*lo g(x + (a/b)^(1/3))/(b^3*c*(a/b)^(2/3) - a*b^2*d*(a/b)^(2/3)) - 1/3*c^2*log (x + (c/d)^(1/3))/(b*c*d^2*(c/d)^(2/3) - a*d^3*(c/d)^(2/3)) + x/(b*d)
Time = 0.29 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {a^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b^{2} c - a^{2} b d\right )}} + \frac {c^{2} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} d - a c d^{2}\right )}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3} c - \sqrt {3} a b^{2} d} - \frac {\left (-c d^{2}\right )^{\frac {1}{3}} c \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d^{2} - \sqrt {3} a d^{3}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} a \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c - a b^{2} d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {1}{3}} c \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{2} - a d^{3}\right )}} + \frac {x}{b d} \]
-1/3*a^2*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2*c - a^2*b*d) + 1/3 *c^2*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^2*d - a*c*d^2) + (-a*b^2 )^(1/3)*a*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b ^3*c - sqrt(3)*a*b^2*d) - (-c*d^2)^(1/3)*c*arctan(1/3*sqrt(3)*(2*x + (-c/d )^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c*d^2 - sqrt(3)*a*d^3) + 1/6*(-a*b^2)^(1 /3)*a*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(b^3*c - a*b^2*d) - 1/6*(-c *d^2)^(1/3)*c*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c*d^2 - a*d^3) + x/(b*d)
Time = 1.93 (sec) , antiderivative size = 873, normalized size of antiderivative = 2.95 \[ \int \frac {x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\ln \left (a\,x+b^2\,c\,{\left (-\frac {a^4}{b^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-a\,b\,d\,{\left (-\frac {a^4}{b^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (\frac {a^4}{-27\,a^3\,b^4\,d^3+81\,a^2\,b^5\,c\,d^2-81\,a\,b^6\,c^2\,d+27\,b^7\,c^3}\right )}^{1/3}+\ln \left (c\,x+a\,d^2\,{\left (\frac {c^4}{d^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-b\,c\,d\,{\left (\frac {c^4}{d^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (\frac {c^4}{27\,a^3\,d^7-81\,a^2\,b\,c\,d^6+81\,a\,b^2\,c^2\,d^5-27\,b^3\,c^3\,d^4}\right )}^{1/3}+\frac {x}{b\,d}+\frac {\ln \left (\frac {3\,x\,\left (a^6\,c^2\,d^4+a^2\,b^4\,c^6\right )}{b\,d}-\frac {3\,a\,c^2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a^4}{b^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (a^5\,d^5-a^4\,b\,c\,d^4+a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,d}\right )\,{\left (\frac {a^4}{-27\,a^3\,b^4\,d^3+81\,a^2\,b^5\,c\,d^2-81\,a\,b^6\,c^2\,d+27\,b^7\,c^3}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (\frac {3\,x\,\left (a^6\,c^2\,d^4+a^2\,b^4\,c^6\right )}{b\,d}+\frac {3\,a\,c^2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a^4}{b^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (a^5\,d^5-a^4\,b\,c\,d^4+a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,d}\right )\,{\left (\frac {a^4}{-27\,a^3\,b^4\,d^3+81\,a^2\,b^5\,c\,d^2-81\,a\,b^6\,c^2\,d+27\,b^7\,c^3}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (\frac {3\,x\,\left (a^6\,c^2\,d^4+a^2\,b^4\,c^6\right )}{b\,d}+\frac {3\,a^2\,c\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {c^4}{d^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (a^5\,d^5-a^4\,b\,c\,d^4+a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,b}\right )\,{\left (\frac {c^4}{27\,a^3\,d^7-81\,a^2\,b\,c\,d^6+81\,a\,b^2\,c^2\,d^5-27\,b^3\,c^3\,d^4}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (\frac {3\,x\,\left (a^6\,c^2\,d^4+a^2\,b^4\,c^6\right )}{b\,d}-\frac {3\,a^2\,c\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {c^4}{d^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (a^5\,d^5-a^4\,b\,c\,d^4+a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,b}\right )\,{\left (\frac {c^4}{27\,a^3\,d^7-81\,a^2\,b\,c\,d^6+81\,a\,b^2\,c^2\,d^5-27\,b^3\,c^3\,d^4}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
log(a*x + b^2*c*(-a^4/(b^4*(a*d - b*c)^3))^(1/3) - a*b*d*(-a^4/(b^4*(a*d - b*c)^3))^(1/3))*(a^4/(27*b^7*c^3 - 27*a^3*b^4*d^3 + 81*a^2*b^5*c*d^2 - 81 *a*b^6*c^2*d))^(1/3) + log(c*x + a*d^2*(c^4/(d^4*(a*d - b*c)^3))^(1/3) - b *c*d*(c^4/(d^4*(a*d - b*c)^3))^(1/3))*(c^4/(27*a^3*d^7 - 27*b^3*c^3*d^4 + 81*a*b^2*c^2*d^5 - 81*a^2*b*c*d^6))^(1/3) + x/(b*d) + (log((3*x*(a^2*b^4*c ^6 + a^6*c^2*d^4))/(b*d) - (3*a*c^2*(3^(1/2)*1i - 1)*(-a^4/(b^4*(a*d - b*c )^3))^(1/3)*(a^5*d^5 - b^5*c^5 + a*b^4*c^4*d - a^4*b*c*d^4))/(2*d))*(a^4/( 27*b^7*c^3 - 27*a^3*b^4*d^3 + 81*a^2*b^5*c*d^2 - 81*a*b^6*c^2*d))^(1/3)*(3 ^(1/2)*1i - 1))/2 - (log((3*x*(a^2*b^4*c^6 + a^6*c^2*d^4))/(b*d) + (3*a*c^ 2*(3^(1/2)*1i + 1)*(-a^4/(b^4*(a*d - b*c)^3))^(1/3)*(a^5*d^5 - b^5*c^5 + a *b^4*c^4*d - a^4*b*c*d^4))/(2*d))*(a^4/(27*b^7*c^3 - 27*a^3*b^4*d^3 + 81*a ^2*b^5*c*d^2 - 81*a*b^6*c^2*d))^(1/3)*(3^(1/2)*1i + 1))/2 + (log((3*x*(a^2 *b^4*c^6 + a^6*c^2*d^4))/(b*d) + (3*a^2*c*(3^(1/2)*1i - 1)*(c^4/(d^4*(a*d - b*c)^3))^(1/3)*(a^5*d^5 - b^5*c^5 + a*b^4*c^4*d - a^4*b*c*d^4))/(2*b))*( c^4/(27*a^3*d^7 - 27*b^3*c^3*d^4 + 81*a*b^2*c^2*d^5 - 81*a^2*b*c*d^6))^(1/ 3)*(3^(1/2)*1i - 1))/2 - (log((3*x*(a^2*b^4*c^6 + a^6*c^2*d^4))/(b*d) - (3 *a^2*c*(3^(1/2)*1i + 1)*(c^4/(d^4*(a*d - b*c)^3))^(1/3)*(a^5*d^5 - b^5*c^5 + a*b^4*c^4*d - a^4*b*c*d^4))/(2*b))*(c^4/(27*a^3*d^7 - 27*b^3*c^3*d^4 + 81*a*b^2*c^2*d^5 - 81*a^2*b*c*d^6))^(1/3)*(3^(1/2)*1i + 1))/2